Using differentials, find the approximate values of the following:
(0.009)1/3
Let us assume that
Also, let x = 0.008 so that x + Δx = 0.009
⇒ 0.008 + Δx = 0.009
∴ Δx = 0.001
On differentiating f(x) with respect to x, we get
We know
When x = 0.008, we have
Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as
Here, and Δx = 0.001
⇒ Δf = (8.3333)(0.001)
∴ Δf = 0.0083333
Now, we have f(0.009) = f(0.008) + Δf
⇒ f(0.009) = 0.2 + 0.0083333
∴ f(0.009) = 0.2083333
Thus, (0.009)1/3 ≈ 0.2083333